Spectral stability of inviscid columnar vortices
Thierry Gallay Institut Fourier
Universit´e Grenoble Alpes, CNRS 100 rue des Maths
38610 Gi`eres, France
Didier Smets
Laboratoire Jacques-Louis Lions Sorbonne Universit´e
4, place Jussieu 75005 Paris, France
June 6, 2019
Dedicated to the memory of Louis N. Howard
Abstract
Columnar vortices are stationary solutions of the three-dimensional Euler equations with axial symmetry, where the velocity field only depends on the distance to the axis and has no component in the axial direction. Stability of such flows was first investigated by Lord Kelvin in 1880, but despite a long history the only analytical results available so far provide necessary conditions for instability under either planar or axisymmetric perturbations. The purpose of this paper is to show that columnar vortices are spectrally stable with respect to three-dimensional perturbations with no particular symmetry. Our result applies to a large family of velocity profiles, including the most common models in atmospheric flows and engineering applications. The proof is based on a homotopy argument, which allows us to concentrate in the spectral analysis of the linearized operator to a small neighborhood of the imaginary axis, where unstable eigenvalues can be excluded using integral identities and a careful study of the so-called critical layers.
1 Introduction
An interesting open question in hydrodynamic stability theory is whether the balance between the centrifugal force and the pressure gradient in axisymmetric vortex flows may lead to an in- stability even if the vorticity profile is monotone and the velocity field has no axial component.
For incompressible perfect fluids, partial answers have been obtained under additional symme- try assumptions. For instance, in the restricted framework of two-dimensional flows, radially symmetric vortices are known to be stable if the vorticity distribution is a monotone function of the distance to the vortex center [20, 19], but even in that idealized situation no sharp stability criterion seems to be available. In the three-dimensional case, the simplest vortex-like equilibria are columnar vortices, namely axisymmetric flows with no vertical velocity and no dependence upon the vertical coordinate. In such flows, all streamlines are horizontal circles centered on the vertical symmetry axis. According to a celebrated result of Rayleigh [21], columnar vortices are stable with respect to axisymmetric perturbations if the square of the velocity circulation along the streamlines is a nondecreasing function of the distance to the symmetry axis, and that condition is actually sharp [24].
A natural question arises from these centennial results: When the vorticity profile is mono- tone and Rayleigh’s condition is satisfied, are columnar vortices stable against three-dimensional
perturbations with no particular symmetry ? Although instabilities have never been observed experimentally or numerically for such vortices in the absence of axial flow, we could not find in the literature even a plausible formal argument supporting the affirmative answer, see Sec- tion 1.3 below for a short historical discussion. In the present paper, we give a rigorous proof of spectral stability for a large family of inviscid columnar vortices without imposing any sym- metry assumption on the class of allowed perturbations. We thus provide an answer to an important question that dates back to the pioneering work of Lord Kelvin [27], who was the first to investigate the three-dimensional stability of vortex columns.
Before stating our results, we first describe the precise framework. We start from the incom- pressible Euler equation in the whole space R3:
∂tu+ (u· ∇)u = −∇p , divu = 0, (1.1) where u=u(x, t) ∈R3 denotes the fluid velocity andp=p(x, t)∈Rthe internal pressure. We mainly consider the vorticity ω(x, t) = curlu(x, t), which describes the local rotation of the fluid particles. Since we are interested in the stability of axially symmetric flows, it is convenient to use cylindrical coordinates (r, θ, z) defined byx1 =rcosθ,x2 =rsinθ, andx3 =z. The velocity and vorticity fields are then decomposed as follows :
u = ur(r, θ, z, t)er+uθ(r, θ, z, t)eθ+uz(r, θ, z, t)ez, ω = ωr(r, θ, z, t)er+ωθ(r, θ, z, t)eθ+ωz(r, θ, z, t)ez,
where er, eθ, ez are unit vectors in the radial, azimuthal, and vertical directions, respectively.
In these coordinates, the vorticity equation∂tω+ (u· ∇)ω−(ω· ∇)u= 0 becomes
∂tωr+ (u· ∇)ωr−(ω· ∇)ur = 0,
∂tωθ+ (u· ∇)ωθ−(ω· ∇)uθ = 1r urωθ−uθωr ,
∂tωz+ (u· ∇)ωz−(ω· ∇)uz = 0,
(1.2)
whereu· ∇=ur∂r+1ruθ∂θ+uz∂z andω· ∇=ωr∂r+1rωθ∂θ+ωz∂z. The velocity field satisfies the incompressibility condition
1
r∂r(rur) +1
r∂θuθ+∂zuz = 0, (1.3)
and can be expressed in terms of the vorticity by solving the linear elliptic system 1
r∂θuz−∂zuθ = ωr, ∂zur−∂ruz = ωθ, 1
r∂r(ruθ)− 1
r∂θur = ωz. (1.4) 1.1 Columnar vortices
Columnar vortices are stationary solutions of (1.1)–(1.4) of the particular form
u = V(r)eθ, ω = W(r)ez, p = P(r), (1.5) whereV is the velocity profile andW the vorticity distribution. The pressureP inside the vortex is determined, up to an irrelevant additive constant, by the centrifugal balance rP′(r) =V(r)2. Instead of V, we prefer using the angular velocity Ω(r) =V(r)/r, which has the same physical dimension as the vorticity W. As a consequence of (1.4), we have
W(r) = 1
r∂r rV(r)
= rΩ′(r) + 2Ω(r). (1.6)
Here are typical examples that are often considered in the literature : 1. The Rankine vortex :
Ω(r) =
(1 if r≤1,
r−2 if r≥1, W(r) =
(2 if r <1,
0 if r >1. (1.7)
As is clear from (1.7), the flow of Rankine’s vortex corresponds to a rigid rotation forr <1 and an irrotational motion forr >1. Although non-physical because of the singularity atr= 1, this flow is relatively easy to analyze mathematically due to the very simple form of the vorticity distribution W, which is a piecewise constant function. The dynamical stability of Rankine’s vortex was first investigated by L. Kelvin as early as 1880 [27].
2. The Kaufmann-Scully vortex :
Ω(r) = 1
1 +r2 , W(r) = 2
(1 +r2)2 , r >0. (1.8) This smooth vortex is characterized by a relatively slow decay of the vorticity distribution as r→ ∞. It has also a very simple analytical form, and is often used as a model for vortices that appear in atmospheric flows or in laboratory experiments, see e.g. [2, Section 3.3.4].
3. The Lamb-Oseen vortex :
Ω(r) = 1 r2
1−e−r2
, W(r) = 2e−r2, r >0. (1.9) Among all solutions of the form (1.5), the Lamb-Oseen vortex plays a distinguished role in connection with the long-time asymptotics of viscous planar flows. Indeed, if viscosity is taken into account, it is known that all localized distributions of vorticity evolve toward a Gaussian vorticity profile as t→ +∞, see [10]. In particular, the Lamb-Oseen vortex is the only one in the above family which corresponds to a self-similar solution of the Navier-Stokes equations.
Remark 1.1. Throughout this paper, it is understood that all independent and dependent variables in the Euler equations (1.1) are dimensionless. Examples (1.7)–(1.9) are normalized so that the vortex core has a diameter of sizeO(1), but that choice can be modified by a simple rescaling. Also, we assume without loss of generality that all vortices are normalized so that Ω(0) = 1, which impliesW(0) = 2.
To study the dynamical stability of the columnar vortex (1.5), we look for solutions of (1.2), (1.3) of the form
u(r, θ, z, t) = rΩ(r)eθ+ ˜u(r, θ, z, t), ω(r, θ, z, t) = W(r)ez+ ˜ω(r, θ, z, t),
where Ω = V /r is the angular velocity of the vortex andW the vorticity distribution given by (1.6). Inserting this Ansatz into (1.2), neglecting the quadratic terms in ˜u and ˜ω, and finally dropping the tildes for notational simplicity, we arrive at the linearized evolution equations
∂tωr+ Ω(r)∂θωr = W(r)∂zur,
∂tωθ+ Ω(r)∂θωθ = W(r)∂zuθ+rΩ′(r)ωr, (1.10)
∂tωz+ Ω(r)∂θωz = W(r)∂zuz−W′(r)ur,
which are the starting point of our analysis. Of course, the linear relations (1.3), (1.4) still hold for the perturbed velocity and vorticity.
It is a classical observation that equations (1.10) can be considered as a self-contained evo- lution system for the vorticity ω, provided the velocity u is expressed in terms of ω by solving the linear elliptic system (1.3), (1.4). Once this is done, we can rewrite (1.10) in the compact form
∂tω = Lω , (1.11)
whereLis a vector-valued, nonlocal, first order differential operator. Our purpose is to study the spectral properties of that operator, and to show thatLhas no spectrum outside the imaginary axis under general assumptions on the angular velocity Ω or the vorticity distribution W.
Another fundamental remark is that system (1.2)–(1.4) is invariant under rotations about the vertical axis, and under translations along that axis. Using a Fourier series expansion with respect to the angular variable θ and a Fourier transform in the vertical variablez, we are led to consider velocities and vorticities of the following particular form
u(r, θ, z, t) = um,k(r, t)eimθeikz, ω(r, θ, z, t) = ωm,k(r, t)eimθeikz, (1.12) where m ∈ Z is the angular Fourier mode and k ∈ R is the vertical wave number. Here u, ω are complex-valued functions, but we impose that um,k =u−m,−k and ωm,k =ω−m,−k so as to obtain real functions after summing over all possible values of m, k. Dropping the subscripts m, k for notational simplicity, we see that the perturbation equations (1.10) translate into
(∂t+imΩ(r))ωr = W(r)ikur,
(∂t+imΩ(r))ωθ = W(r)ikuθ+rΩ′(r)ωr, (1.13) (∂t+imΩ(r))ωz = W(r)ikuz−W′(r)ur.
In addition, the following relations hold : ωr = imr uz−ikuθ,
ωθ = ikur−∂ruz, ωz = 1r∂r(ruθ)−imr ur,
and 1
r∂r(rur) +im
r uθ+ikuz = 0. (1.14) As before, we can rewrite (1.13) in the compact form
∂tω = Lm,kω , (1.15)
assuming that the velocity u=um,k in (1.13) is expressed in terms of the vorticity ω=ωm,k by solving the linear relations (1.14) with appropriate boundary conditions. The main properties of the Biot-Savart mapωm,k 7→um,k obtained in this way will be recalled in Section 6.1. Being an integral operator acting on functions of the sole variabler, the generatorLm,k in (1.15) is of course easier to study than the original three-dimensional differential operatorL in (1.11).
1.2 Statement of the results
To state our results in a precise way, we first specify our hypotheses on the unperturbed columnar vortex. We find it convenient to formulate these assumptions at the level of thevorticity profile W. Note that, in view of (1.6), the angular velocity Ω can be expressed in terms of W by the formula
Ω(r) = 1 r2
Z r 0
W(s)sds , r >0, (1.16)
and the derivative of Ω is in turn given by Ω′(r) = W(r)−2Ω(r)
r = 1
r3 Z r
0
W′(s)s2ds , r >0. (1.17)
In what follows, we denoteR+= (0,∞) and R+ = [0,∞).
Assumption H1: The vorticity profile W : R+ → R+ is a C1 function satisfying W′(0) = 0, W′(r)<0 for all r >0, and the total circulation
2πΓ = 2π Z ∞
0
W(r)rdr (1.18)
of the columnar vortex is finite.
Under assumption H1 the angular velocity profile Ω ∈ C1(R+)∩ C2(R+) given by (1.16) is positive and satisfies Ω(0) = W(0)/2, Ω′(0) = 0, Ω′(r) < 0 for all r >0, and Ω(r) ∼ Γ/r2 as r→ ∞. In particular, theRayleigh function Φ : [0,∞)→Rdefined by
Φ(r) = 2Ω(r)W(r), r ≥0, (1.19)
is positive everywhere. As a matter of fact, in our framework assumption H1 corresponds exactly to the combination of Rayleigh’s condition [21] and of the two-dimensional stability criterion [20, 19]. We supplement it with the following:
Assumption H2: TheC1 functionJ :R+→R+ defined by J(r) = Φ(r)
Ω′(r)2 , r >0, (1.20)
satisfies J′(r)<0 for allr >0 and rJ′(r)→ 0 asr → ∞.
This second assumption is more technical in nature, and certainly more difficult to justify. We first observe that it is satisfied for the Kaufmann-Scully vortex (1.8), becauseJ(r) = 1 + 1/r2 in that case, and a direct calculation that can be found in Section 6.7 below reveals that assumption H2 also holds for the Lamb-Oseen vortex (1.9). A quantity corresponding to (1.20) appears in the work of G.I. Taylor [26] on the stability of stratified shear flows; in that context it is called thelocal Richardson number (see e.g. [7, Chapter 6]). Its relevance for stability was confirmed by Miles [18] and Howard [13]. The ideas of Howard were translated into the columnar vortex framework by Howard and Gupta [14], where the quantity (1.20) is also shown to play an important role in the stability analysis for perturbations with nonzero angular Fourier modem and nonzero vertical wave numberk. Indeed, it is proved in [14] that the linear operator Lm,k in (1.15) has no unstable eigenvalue if
k2
m2 J(r)≥ 1
4, for all r >0, (1.21)
see also Proposition 3.4 below. Note that, in the case of the Lamb-Oseen vortex, inequality (1.21) is always violated for large r > 0 because J(r) → 0 as r → ∞, whereas (1.21) holds for the Kaufmann-Scully vortex if and only ifm2≤4k2. Although Howard and Gupta’s result alone is not sufficient, it plays a crucial role in our stability analysis in Section 4, where we have to distinguish two spatial regions according to whether the local Richardson number (k2/m2)J(r) is greater or smaller than 1/4. It turns out to be important for our approach that inequality (1.21) either holds for all r ≥0, or is satisfied if and only if r ≤r∗ for some r∗ >0. The only way to enforce that property for all possible values of m and k is to assume that the function J in (1.20) is decreasing. However, there is no evidence that assumption H2 is more than a technical limitation, and we hope that this question will be clarified in the future.
Remark 1.2. Although this is not immediately obvious, assumption H2 implies the existence of a nonnegative number ℓ∞≥0 such that
r→∞lim r4W(r) = ℓ∞, lim
r→∞r5W′(r) = −4ℓ∞, (1.22) see Section 6.4 below.
Next, we specify the function space in which we study the linearized operator Lm,k defined in (1.13), (1.15). Since we used a Fourier decomposition to reduce our analysis to functions of the form (1.12), it is natural to work in L2-based function spaces. Given m∈Zand k∈R, we thus define the enstrophy space
Xm,k = n
ω∈L2(R+, rdr)3
1
r∂r(rωr) +im
r ωθ+ikωz = 0o
, (1.23)
equipped with the norm kωk2L2 =
Z ∞
0 |ω(r)|2rdr , where |ω|2 = |ωr|2+|ωθ|2+|ωz|2.
It is not difficult to verify that the generator Lm,k of the linearized evolution equation (1.15) defines a bounded linear operator in the space Xm,k if k6= 0, see Proposition 2.1 below. With this observation in mind, we can formulate our first main result :
Theorem 1.3. Consider a columnar vortex whose vorticity profile W satisfies assumptions H1, H2 above. Given m∈Z and k6= 0, let Lm,k be the generator of the linearized evolution (1.15).
Then the spectrum of Lm,k in the enstrophy space Xm,k satisfies
σ(Lm,k) ⊂ iR. (1.24)
Remark 1.4. The proof actually shows that, under the normalization condition W(0) = 2, σ(Lm,k) consists of essential spectrum filling the closed interval {−imb|0 ≤ b ≤ 1} ⊂ iR, and of a countable family of simple, purely imaginary eigenvalues that accumulate only at
−im∈iR. These eigenvalues are well studied in the physical literature (a brief account is given in Section 1.3 below), and the corresponding eigenfunctions are referred to as Kelvin vibration modes. The main contribution of the present paper is to show that the operator Lm,k has no eigenvalue outside the imaginary axis, if the vorticity profileW satisfies assumptions H1, H2. It is interesting to note that this result remains valid for the Rankine vortex (1.7) which does not satisfy our hypotheses, see Section 6.2 below.
Remark 1.5. The particular case k= 0, which corresponds to two-dimensional perturbations, is excluded in Theorem 1.3 because the function spaceXm,k is not appropriate in that situation.
This is essentially due to the fact that the two-dimensional Biot-Savart law is ill-defined for vorticities in the enstrophy space. The problem can be eliminated by introducing a radial weight that ensures a faster decay of ω(r) as r → ∞, or alternatively by working in the energy space as mentioned in Remark 1.7 below. However, since the two-dimensional stability of radially symmetric vortices is already well documented, we chose to ignore these technical issues and to concentrate here on the genuinely three-dimensional casek6= 0, which was essentially unexplored until now.
According to Theorem 1.3, for anys∈Cwith Re(s)6= 0, the resolvent operator (s−Lm,k)−1 is well defined and bounded in the spaceXm,k if m∈Zandk6= 0. Actually, one can prove that the resolvent is uniformly bounded for all m ∈ Zand for all nonzero k in the one-dimensional lattice Zk0, where k0 > 0 is arbitrary. Returning to the full linearized evolution (1.11), this proves spectral stability of the generatorL in the space
L˙2σ,per,h = n
ω∈L2(R2×Th)3
divω = 0, Z h
0
ω(x1, x2, x3) dx3 = 0o
, (1.25) where Th =R/(Zh) and h= 2π/k0 is the vertical period. We can thus state our second main result :
Theorem 1.6. Under the assumptions of Theorem 1.3, let Ldenote the full linearized operator in (1.11). Then, for any h >0, the spectrum of L in the space L˙2σ,per,h satisfies
σ(L) = iR. (1.26)
Remark 1.7. The reason for restricting ourselves to functions with zero average in the vertical direction was explained in Remark 1.5. The same technical limitation prevents us from consid- ering perturbations in the enstrophy space L2σ(R3), without assuming periodicity in the vertical direction, because in that case all values of the vertical wave numberk∈Rhave to be taken into account. In a subsequent work [11], we use Theorem 1.3 to obtain the equivalent of Theorem 1.6 for the Euler equation in velocity formulation. There we consider perturbations in the energy space, and we also obtain semigroup estimates for the linearized operator at a columnar vortex.
In the proof of Theorems 1.3 and 1.6, we find it convenient to normalize our velocity and vorticity profiles so that Ω(0) = 1 and W(0) = 2. This leads to the following definition :
Definition 1.8. We denote by W the class of all vorticity profiles W :R+→R+ satisfying the assumptions H1, H2 above, as well as the normalizing condition W(0) = 2.
It is worth emphasizing here that assumption H2 involves the function J defined in (1.20), which depends nonlinearly on the vorticity profileW. As a consequence, our family of admissible profiles is not a vector space, and the classW introduced in Definition 1.8 is not even a convex set. However, we shall prove in Section 6.4 that any profile W ∈W is entirely determined by the auxiliary function
Q(r) = 1
p1 +J(r), r >0, (1.27)
and that the class W can be described by simple linear constraints at the level of the function Q. This makes it possible to perform continuous interpolation and approximation within the classW, and such tools will play a crucial role in the proof of Theorem 1.3.
Remark 1.9. If we equip the class W with the topology of Cb1(R+), the Banach space of all bounded continuously differentiable functions onR+with bounded derivative, it is easily verified that the linear operator operatorLm,k ∈ L(Xm,k) depends continuously on the vorticity profile W ∈W, see Lemma 4.1 below. In particular, isolated eigenvalues ofLm,k outside the imaginary axis (if they are any) vary continuously when W is perturbed in that topology. This implies that the conclusion (1.24) of Theorem 1.3 remains valid for any vorticity profile that belongs to the closure of the class W in Cb1(R+). This larger class contains vorticities W that are not strictly decreasing functions of the radiusr, and may even be compactly supported.
1.3 Previous results and perspectives
The first historical contribution regarding the stability of columnar vortices in incompressible fluids is of course the seminal work [27] by Kelvin. In that study, the focus is put on neutral modes, namely eigenmodes of the linearized Euler equation that correspond to purely imaginary eigenvalues; these were later termed “Kelvin vibration modes”. As Kelvin expresses it: “The problem thus solved is the finding of the periodic disturbance in the motion of rotating liquid [...]”. The computations in [27] are performed in situations where the underlying axisymmetric flow has piecewise constant vorticity; this exactly corresponds to what was called the Rankine vortex in Section 1.1 above. However, Kelvin waves are observed to play an important role in the dynamics of the Euler equation for a much wider variety of profiles, and were actively studied in the literature since then (in most cases numerically, or using asymptotic expansions combined
with physical arguments). In the case of the Lamb-Oseen vortex, important contributions were made in particular by Le Diz`es and Lacaze [16] and Fabre, Sipp and Jacquin [9], both in the inviscid case and in the vanishing viscosity limit. Unlike Kelvin (who had no computer account!), the authors of [16, 9] also consider the possibility of eigenvalues off the imaginary axis. One of the conclusions of [9] based on their numerical findings is that “[...]no amplified modes were found, a result which demonstrates the stability of the Lamb-Oseen vortex.”
In a different direction, Rayleigh [20, 21] initiated the study of necessary conditions for columnar vortex instability1. Although it may certainly be found physically convincing, the original argument [21] leading to Rayleigh’s criterion cannot be easily transposed into rigorous mathematical terms. Instead, the approach followed by Howard and Gupta [14], which we consider one of the most interesting and important contributions so far, is both rigorous and elementary. This remarkable work contains most importantly a non-conclusive but enlightening section called “Remarks on the non-axisymmetric case”, in which the partial stability criterion (1.21) can be found. The authors write: “The overall conclusion of this consideration of the non- axisymmetric case is thus essentially negative: the methods used to derive the Richardson number and semicircle results in the axisymmetric case reproduce the known results of Rayleigh for two- dimensional perturbations and pure axial flow, but seem to give very little more. In fact the present situation with regard to non-axisymmetric perturbations seems to be very unsatisfactory from a theoretical point of view.”
Attempts have been made to derive necessary conditions for instability extending Rayleigh’s criterion to non-axisymmetric perturbations. One such criterion was proposed by Billant and Gallaire [5], following earlier work by Leibovich and Stewartson [17], and applies in a given Fourier sector. It is relatively simple to state but requires a number of a posteriori checks which could be more difficult to perform. As the authors mention, in all the situations they tested the most unstable modes were always the axisymmetric ones (this is reminiscent of Squire’s theorem in the context of viscous shear flows), and therefore, in practice, Rayleigh’s criterion appears to be sufficient to detect potential instabilities. Yet, a priori estimates on the possible growth in a given Fourier sector are certainly interesting per se.
Spectral stability of course does not imply stability of the flow for a Hamiltonian system such as Eq. (1.1). In a celebrated paper [3, 4], Arnold derived a nonlinear stability criterion for stationary solutions of the Euler equations, which are viewed as critical points of the kinetic energy functional over the manifold of isovortical vector fields, and he treated in detail the case of 2D flows. His approach was subsequently extended by Szeri and Holmes [25] and applied to axisymmetric perturbations of columnar vortices. A few years later, Rouchon [22] proved that the conditions in Arnold’s criterion are never satisfied if one considers genuinely 3D perturbations of nontrivial stationary flows. An intermediate step between spectral and nonlinear stability is linear stability, which consists in controling the growth of the semigroup generated by the linearized operator in Theorem 1.6. Preliminary results in that direction can be found in the subsequent work [11].
We close this section mentioning that a number of interesting phenomena are known to arise, as far as instabilities are concerned, when the base flow possesses an additional axial component.
Some of the works already quoted, and many others, do consider that situation as well. Since we did not investigate it at all in this work, we keep that discussion for another occasion.
1Or equivalently sufficient conditions for their stability; in the present work stability is only understood in the spectral sense, meaning the absence of eigenvalues with positive real part.
1.4 Organization of the paper
Our strategy to prove Theorems 1.3 and 1.6 can be explained as follows. In a first step, we show in Section 2 that the essential spectrum of the operatorLm,k is purely imaginary. The rest of the spectrum consists of isolated eigenvalues with finite multiplicity, and the corresponding eigenfunctions are solutions of a second order differential equation involving a complex potential that depends on m, k, and the spectral parameter s. The eigenvalue equation is difficult to study in general, but using techniques that date back to Rayleigh [20, 21] it is easy to verify that it has no nontrivial solution with Re(s)6= 0 when the perturbations are either axisymmetric (m= 0) or two-dimensional (k= 0). In Section 3, we establish a few preliminary results in the case werem6= 0 and k6= 0. In particular, we derive useful identities satisfied by any nontrivial eigenfunction, and we recover the stability criterion (1.21) of Howard and Gupta. The core of the proof of Theorem 1.3 is Section 4. We construct a suitable homotopy between the vorticity profile W ∈ W and a reference profile for which stability in the corresponding Fourier sector Xm,k is known by Howard and Gupta’s criterion. By a continuity argument, this strategy allows us to reduce the problem to proving the absence of unstable eigenvaluesarbitrarily close to the imaginary axis, for a one-parameter family of profiles in the classW. A delicate combination of integral identities and comparison arguments relying on assumption H2 are then used to perform such a “critical layer analysis” and hence to preclude the existence of unstable eigenvalues in the large. Finally, in Section 5, we prove uniform resolvent estimates for the linear operator Lm,k outside the imaginary axis, which imply that the full linearization Lhas indeed no spectrum in that region when acting on the space ˙L2σ,per,h for any h > 0. This is precisely the conclusion of Theorem 1.6. The last section is an appendix were several auxiliary results are established.
In particular, we give useful estimates for the Biot-Savart law in the Fourier sector indexed by m, k, we prove the stability of Rankine’s vortex (1.7) which is not covered by Theorem 1.3, and we explain how to perform continuous interpolation and approximation in the nonlinear class W.
Acknowledgements. The authors were partially supported by grants ANR-13-BS01-0003-01 (Th.G.) and ANR-14-CE25-0009-01 (D.S.) from the “Agence Nationale de la Recherche”. They benefited from discussions with S. Le Diz`es, in particular during the meeting “Vortex et solitons pour les fluides classiques et quantiques” (CIRM, Marseille, 2012) where this work was initiated, and also from insightful remarks from an anonymous referee.
2 Formulation of the spectral problem
Let W be a vorticity profile in the class W, and let Ω be the corresponding angular velocity defined by (1.16). For a fixed value of the angular Fourier modem∈Zand of the vertical wave number k∈R, we consider the linear operatorLm,k introduced in (1.15). In view of (1.13), we have the natural decomposition
Lm,k = Am+Bm,k, (2.1)
whereAm is the multiplication operator defined by
Amω = −imΩ(r)ω+rΩ′(r)ωreθ, (2.2) and Bm,k is the following nonlocal perturbation :
Bm,kω = ikW(r)u−W′(r)urez. (2.3) Hereu= (ur, uθ, uz) denotes the velocity obtained from the vorticityω = (ωr, ωθ, ωz) by solving the linear PDE system (1.14) with appropriate boundary conditions. We refer the reader to
Section 6.1 below for a discussion of the map ω 7→ u, which we call the Biot-Savart law in the Fourier subspace indexed by m and k. Our main goal in this paper is to study the spectral properties of the operator Lm,k acting on the enstrophy spaceXm,k defined by (1.23).
The following simple result is the starting point of our analysis.
Proposition 2.1. Fix m∈Z and k∈R\ {0}.
1) The linear operator Am defined by (2.2) is bounded in Xm,k with spectrum given by σ(Am) = n
z∈C
z=−imb for some b∈[0,1]o
. (2.4)
This spectrum is purely continuous if m6= 0, and reduces to a single eigenvalue ifm= 0.
2) The linear operator Bm,k defined by (2.3) is compact in Xm,k.
Proof. Given s ∈ C and f = (fr, fθ, fz) ∈ Xm,k, the resolvent equation (s−Am)ω = f is equivalent to the linear system
(s+imΩ(r))ωr = fr, (s+imΩ(r))ωθ = fθ+rΩ′(r)ωr, (s+imΩ(r))ωz = fz. (2.5) AsW ∈W, we know that Ω : [0,∞) →R+ is strictly decreasing with Ω(0) = 1 and Ω(r)→0 as r→ ∞. Thus, ifs6=−imbfor allb∈[0,1], the quantity|s+imΩ(r)|is bounded away from zero, and it follows that system (2.5) has a unique solution ω ∈Xm,k satisfying kωkL2 ≤C(s)kfkL2. On the other hand, if m 6= 0 and s = −imb for some b ∈ [0,1], it is easy to verify that the operator s−Am is one-to-one but not onto (its range is dense but strictly contained inXm,k), so that sbelongs to the continuous spectrum of Am. Finally, if m= 0, it is clear that s= 0 is an eigenvalue of Am, with infinite multiplicity. This proves the first part.
We next consider the operator Bm,k. If ω ∈ Xm,k and kωkL2 ≤ 1, Proposition 6.1 shows that the associated velocity field u satisfies k∂rukL2 +kkukL2 ≤C for some universal constant C >0. This gives a uniform bound on u inH1(R+, rdr) since we assume that k 6= 0. By the Fr´echet-Kolmogorov theorem, we deduce that the map ω 7→ Bm,kω = ikW(r)u−W′(r)urez is compact in Xm,k, because the functions W and W′ are bounded and converge to zero as r→ ∞.
Proposition 2.1 shows in particular that, for anym∈Zand anyk∈R\{0}, the linearization Lm,k =Am+Bm,k defines a bounded operator in the spaceXm,k. Moreover, asBm,k is compact, theessential spectrumof Lm,k is the same as the (essential) spectrum of Am, namely the closed interval Im = {−imb|0 ≤ b ≤ 1} ⊂ iR, see [8, Theorem I.4.1]. Note that, in the present case, the various definitions of the essential spectrum listed in [8, Section I.4] all coincide. This implies that the spectrum ofLm,koutside the intervalIm entirely consists of isolated eigenvalues with finite multiplicities, which can accumulate only on the essential spectrum. The proof of Theorem 1.3 is thus reduced to showing that all isolated eigenvalues of Lm,k actually lie on the imaginary axis.
Remark 2.2. As the functions Ω, W are real-valued, it is not difficult to verify, using the definitions (2.2), (2.3) and the relations (1.14) between u and ω, that the spectrum of Lm,k in Xm,k has the following symmetries :
σ(Lm,k) = σ(Lm,−k) = −σ(L−m,k), and σ(Lm,k) = −σ(Lm,k). (2.6) The corresponding mappings between eigenspaces are also easy to establish. In particular, the last relation in (2.6) means that the spectrum of σ(Lm,k) is symmetric with respect to the imaginary axis, a property that will be used later on.
As a first step in the proof of Theorem 1.3, we derive an equation for the eigenfunctions of the operatorLm,k corresponding to eigenvalues outside the essential spectrum. In what follows, we thus assume thats∈Cis an isolated eigenvalue ofLm,kwith eigenfunctionω = (ωr, ωθ, ωz)∈ Xm,k, and we denote by u= (ur, uθ, uz) the velocity field associated withω via the Biot-Savart law, see Section 6.1. As in [7], we define
γ(r) = s+imΩ(r), r >0. (2.7)
Sincesdoes not belong to the essential spectrum ofLm,k by assumption, it follows from Propo- sition 2.1 thatγ(r)6= 0 for all r >0.
In view of (1.13), the eigenvalue equation reads γ(r)ωr = ikW(r)ur,
γ(r)ωθ = ikW(r)uθ+rΩ′(r)ωr, (2.8) γ(r)ωz = ikW(r)uz−W′(r)ur,
where rΩ′(r) = W(r)−2Ω(r) by (1.6). If we express the vorticity ω in terms of u using the relations (1.14), we obtain the equivalent system
ikW(r)ur+ikγ(r)uθ−imγ(r)
r uz = 0, (2.9)
ikγ(r)ur−2ikΩ(r)uθ−∂r(γ(r)uz) = 0, (2.10)
W′(r)−imγ(r) r
ur+γ(r)1
r∂r(ruθ)−ikW(r)uz = 0. (2.11) Assuming for the moment thatk6= 0, it is straightforward to verify that the relations (2.9)–(2.11) together imply the incompressibility condition
1
r∂r(rur) +im
r uθ+ikuz = 0. (2.12)
To reduce system (2.9)–(2.12) to a single equation, we first express the azimuthal velocity uθ in terms ofur, uz using (2.9), and replace it into (2.10), (2.12) to obtain the 2×2 system
∂r∗−imW(r) rγ(r)
ur+ik
1 + m2 k2r2
uz = 0, (2.13)
∂r+imW(r) rγ(r)
uz−ik
1 + Φ(r) γ(r)2
ur = 0, (2.14)
where Φ = 2ΩW is the Rayleigh function and∂r∗=∂r+1r. Next, observing that the coefficient of uz in (2.13) does note vanish, we can divide (2.13) by that coefficient and apply the differential operator ∂r + imWrγ to obtain, with the help of (2.14), the following second-order differential equation for the radial velocity :
∂r+ imW(r) rγ(r)
r2 m2+k2r2
∂r∗−imW(r) rγ(r)
ur =
1 + Φ(r) γ(r)2
ur. (2.15) If we expand the product in the left-hand side, we find after straightforward calculations
−∂r
r2∂∗rur m2+k2r2
+
1 + 1 γ(r)2
k2r2Φ(r)
m2+k2r2 + imr
γ(r)∂r W(r) m2+k2r2
ur = 0, (2.16) see also [7, Eq. (15.26)]. This is the desired eigenvalue equation, which will be our main concern in the rest of this paper. It is formulated in terms of the radial velocity ur, which satisfies ur ∈H1(R+, rdr) according to Proposition 6.1. In fact, we also have ur∈Hloc2 (R+) in view of the divergence-free condition (2.12).
Remark 2.3. In the case where k= 0, a much simpler calculation shows that the eigenvalue equation is still given by (2.16) if m 6= 0, although the derivation above is not correct. If k=m= 0, equation (2.16) is of course meaningless, but in that case it is obvious that system (2.8) has no nontrivial solution for s6= 0.
Summarizing the arguments developed so far, the proof of Theorem 1.3 can be reduced to showing that, for all m∈Zand all k∈R\ {0}, the eigenvalue equation (2.16) has no nontrivial solutionur∈H1(R+, rdr) if the spectral parameters∈Csatisfies Re(s)6= 0. This is a difficult task in general, which we postpone to Sections 3 and 4. For the time being, we just mention two important particular cases which are relatively easy to handle.
2.1 The axisymmetric case
In the axisymmetric case m= 0, Proposition 2.1 asserts that the essential spectrum of L0,k is reduced to zero, and therefore away from the origin there may only exist eigenvalues with finite multiplicity. The spectral function (2.7) is constant in that case, and the stability equation (2.16) reduces to
−∂r∂r∗ur+k2
1 +Φ(r) s2
ur = 0. (2.17)
The following classical result dates back to the work of L. Rayleigh [21], and is reproduced here for the reader’s convenience.
Proposition 2.4. Assume that the Rayleigh function Φ is nonnegative. Then the eigenvalue equation (2.17) has no nontrivial solution ur ∈H1(R+, rdr) if Re(s)6= 0.
Proof. According to Remark 2.3, we can suppose thatk 6= 0. Assume thatur ∈H1(R+, rdr) is a nontrivial solution of (2.17) for some s∈C\ {0}. Multiplying both sides of (2.17) byru¯r and integrating the resulting expression over R+, we obtain the useful relation
Z ∞ 0
n|∂r∗ur|2+k2
1 +Φ(r) s2
|ur|2o
rdr = 0. (2.18)
By assumption we have R∞
0 Φ|ur|2rdr >0, because ur is a nontrivial solution of (2.17) and Φ is a nonnegative function with Φ(0)> 0. Thus taking the imaginary part of (2.18) we deduce that Im(s2) = 0, hences∈Ror s∈iR. The first possibility is excluded by taking the real part of (2.18), hence we conclude thats∈iR.
Remark 2.5. Actually it was observed by Synge [24] that the Rayleigh stability criterion Φ≥0 is not only sufficient, but also necessary in the axisymmetric case. Indeed, we know that Φ(0) =W(0)2 >0, and for localized vortices we always have Φ(r)→0 asr → ∞. Now, assume that Φ(¯r)<0 for some ¯r >0, and consider the Schr¨odinger equation
−s2∂r∂r∗ur+k2
s2+ Φ(r)
ur = Eur, r >0, (2.19) in the semiclassical limit where 0 < s ≪ 1. As the potential term s2 + Φ(r) takes negative values near r = ¯r, it is well known that the operator in (2.19) has negative eigenvalues E if s >0 is sufficiently small, see e.g. [23, 12]. In fact, the number of negative eigenvalues increases unboundedly ass→0, and this implies by continuity that Eq. (2.19) withE= 0, or equivalently Eq. (2.17), has a nontrivial solution ur ∈ H1(R+, rdr) for a sequence of values of s > 0 that converges to zero.
We also note that the equivalent of Synge’s observation, but used fors∈iRinstead ofs∈R, implies in contrast that, when the Rayleigh function is nonnegative, the linearized operatorL0,k does possess nonzero eigenvalues on the imaginary axis, which correspond to Kelvin modes.
2.2 The two-dimensional case
Although it is not included in Theorem 1.3, the two-dimensional casek= 0 is worth mentioning too. When m6= 0, the eigenvalue equation (2.16) reduces to
−∂r(r2∂r∗ur) +
m2+imrW′(r) γ(r)
ur = 0. (2.20)
A well-known sufficient condition for stability is that the vorticity profile W be a monotone function, see e.g. [19], but unlike in the axisymmetric case no sharp criterion has been established so far. Again, for the reader’s convenience, we reproduce here the easy argument showing spectral stability if W′ has a constant sign.
Proposition 2.6. Assume that the vorticity profileW is monotone. Then the eigenvalue equa- tion (2.20) has no nontrivial solution ur∈H1(R+, rdr) if Re(s)6= 0.
Proof. Assume that ur ∈ H1(R+, rdr) is a nontrivial solution of (2.20) for some s ∈C with Re(s)6= 0. Multiplying both members of (2.20) by ru¯r and integrating over R+, we obtain the relation
Z ∞ 0
|∂r(rur)|2+
m2+ imrW′(r) γ(r)
|ur|2
rdr = 0. (2.21)
In particular, taking the imaginary part and using (2.7), we find mRe(s)
Z ∞
0
W′(r)
|γ(r)|2|ur|2r2dr = 0,
and since W is monotone we conclude thatur is supported in the set whereW′ vanishes. This is clearly impossible if W is not identically constant, because ur is a nontrivial solution of the second order ODE (2.20). But ifW is a constant, equation (2.21) immediately gives the desired contradiction.
3 The eigenvalue equation for m 6 = 0 and k 6 = 0
In this section we begin our study of the eigenvalue equation (2.16) in the general case where m 6= 0 and k 6= 0. In view of the symmetries (2.6), we can assume without loss of generality that m≥1 and k >0. We write the spectral parameter as s=m(a−ib), wherea, b∈R, and we decompose
γ(r) = s+imΩ(r) = imγ⋆(r), where γ⋆(r) = Ω(r)−b−ia . (3.1) According to Proposition 2.1, the essential spectrum of the operator Lm,k is the set of all s=m(a−ib) such thata= 0 and b∈[0,1]. Outside that set, the functionγ⋆ is bounded away from zero for all r >0 and the eigenvalue equation (2.16) becomes
−∂r A(r)∂r∗ur
+B(r)ur = 0, (3.2)
where∂r∗ =∂r+1r and A(r) = r2
m2+k2r2 , B(r) = 1− k2 m2
A(r)Φ(r) γ⋆(r)2 + r
γ⋆(r)∂r
W(r) m2+k2r2
. (3.3)
3.1 Asymptotic behavior at the origin and at infinity
Our first goal is to determine the asymptotic behavior of the solutions of the complex ODE (3.2) as r → 0 and r → ∞, assuming that a 6= 0 or b /∈ [0,1]. We start with the behavior at the origin. Ifur is a solution of (3.2), we set
ur(r) = 1 r v
log1 r
, r >0,
or equivalently v(x) =e−xur(e−x) forx = log(1/r)∈R. The new function v:R→ Rsatisfies the equation
v′′(x) + 2k2A(e−x)v′(x)− C(x)v(x) = 0, where C(x) = e−2x B(e−x)
A(e−x). (3.4) In view of (3.3) we have A(e−x) = O(e−2x) and C(x) = m2+O(e−2x) +O(e−x|W′(e−x)|) as x → +∞. Thus applying e.g. [6, Theorem 3.8.1], we deduce that equation (3.4) as a unique solution vsuch thatemxv(x)→1 asx→+∞. Returning to the original variables, we conclude that equation (3.2) has a unique solution ur such that r1−mur(r) →1 as r →0. This solution ur and its first derivative u′r depend continuously on the various parameters in (3.2), including the vorticity profile W ∈ Cb1(R+) and the spectral parameter s=m(a−ib) ∈ C, uniformly in r on any bounded interval of the form (0, R). Any linearly independent solution of (3.2) blows up like r−1−m asr →0, and is therefore not square integrable near the origin.
We next study the behavior at infinity. Ifuris a solution of (3.2), we definew(r) =r1/2ur(r) and obtain for wthe equation
w′′(r) +A′(r)
A(r)w′(r)− D(r)w(r) = 0, where D(r) = B(r) A(r) + 3
4r2 − 1 2r
A′(r)
A(r) . (3.5) We have A′(r)/A(r) =O(r−3) andD(r) =k2+O(r−2) asr → ∞, because Remark 1.2 implies that W(r) = O(r−4), W′(r) = O(r−5), and Φ(r) = O(r−6) in that limit. Invoking again [6, Theorem 3.8.1], we deduce that (3.5) has a unique solution wsuch that ekrw(r)→1 asr→ ∞, hence (3.2) has a unique solution satisfying r1/2ekrur(r) →1 as r → ∞. This solution and its first derivative depend continuously on the parameters in (3.2), uniformly on the interval (R,∞) for any R >0. Any linearly independent solution of (3.2) grows like r−1/2ekr asr→ ∞, and is therefore not square integrable.
Summarizing, we have shown:
Lemma 3.1. Ifm6= 0 andk6= 0, any eigenvalue of the linear operatorLm,k ∈ L(Xm,k)outside the essential spectrum (2.4) is necessarily simple. Moreover, ifur is the radial velocity profile of the corresponding eigenfunction, there exist α, β∈C such that
r→0limr1−|m|ur(r) = α , and lim
r→∞r1/2e|k|rur(r) = β . 3.2 Eigenvalues on the imaginary axis: Kelvin waves
In a second step, we consider the eigenvalues of the linearized operator Lm,k on the imaginary axis. The corresponding eigenfunctions describe “vibration modes” of the columnar vortex, and were first studied by Kelvin [27] in the particular case of Rankine’s vortex. Strictly speaking, this subsection is not part of the proof of Theorem 1.3, but in view of the physical relevance of the Kelvin waves it is worth mentioning a few results that can be rigorously established.
In what follows, we thus assume thata= 0 andb /∈(0,1), so that γ⋆(r)6= 0 for all r >0. In that case equation (3.2) has real coefficients, and its solutions can be studied using standard ODE techniques. For simplicity we suppose here that the vorticity profile W ∈ W is the restriction to R+ of a smooth even function onRsatisfyingW′′(0)<0, as it is the case for the Kaufmann- Scully vortex (1.8) or the Lamb-Oseen vortex (1.9). We consider separately the regimes where b≥1 andb≤0.
Lemma 3.2. For any m6= 0andk6= 0, the set of allb >1 such that Eq.(3.2) witha= 0has a nontrivial solution inH1(R+, rdr)is a countable family which accumulates only at1. Moreover, Eq. (3.2) has no nontrivial solution in H1(R+, rdr) ifa= 0 and b= 1.
Proof. When b > 1, we apply to Eq. (3.2) the change of variables ur = rmA(r)−1/2v, where A(r) is as in (3.3). A direct calculation shows that the new functionv satisfies
−∂r2v−2m+1
r ∂rv+
k2+F(r) +G(r)
v = 0, r >0, (3.6) where
F(r) = k2A(r) r2
−2 + 3k2A(r)
, G(r) = −k2 m2
Φ(r)
γ⋆(r)2 + r A(r)γ⋆(r)∂r
W(r) m2+k2r2
.
We assume thatb= 1 +h2 for some small h >0, and we expand
−γ⋆(r) = 1 +h2−Ω(r) = h2+ρr2+O(r4), asr→0,
whereρ=−Ω′′(0)/2 =−W′′(0)/8>0. Ifr =hs, it is straightforward to verify that h4
k2+F(hs) +G(hs)
= −4k2 m2
1
(1 +ρs2)2 +O(h2), ash→0,
uniformly for all s >0. Thus the new function w defined by setting w(s) = v(hs) satisfies the semi-classical Schr¨odinger equation
Lhw := −h2
∂s2w+2m+1 s ∂sw
−4k2 m2
w
(1 +ρs2)2 +U(s, h)w = 0, (3.7) for all s >0, where U(s, h) = O(h2) as h → 0, uniformly ins. Since the principal part of the potential term in (3.7) is negative, standard results in semiclassical analysis [23, 12] show that the operatorLh has negative eigenvalues ifhis sufficiently small, and that the number of these bound states is O(h−1) as h → 0. Moreover, as F(r) +G(r) → 0 as r → ∞, the bottom of the essential spectrum of Lh is k2h4 >0 for any h >0. These two observations together imply that Lh has a zero eigenvalue for a countable sequence hn → 0, and returning to the original variables we conclude that Eq. (3.2) with a= 0 has a nontrivial solution in H1(R+, rdr) for a sequence bn= 1 +h2n→1.
When b= 1, namelyh= 0, the leading term in the function B(r)/A(r) satisfies k2
m2 Φ(r)
γ⋆(r)2 = Θ2
r4 1 +O(r2)
asr→0, where Θ2 = 4k2 m2ρ2 .
To investigate the behavior of the solutions of (3.2) near r = 0 in that case, it is useful make the change of variables ur(r) =r−1/2U(1/r). Setting x = 1/r, this leads to an equation of the form
U′′(x) + ˜C(1/x)U′(x) + ˜D(1/x)U(x) = 0, x >0, (3.8)
where ˜C(r) = O(r3) and ˜D(r) = Θ2+O(r2) as r → 0. Using [6, Theorem 3.8.1], we deduce that Eq. (3.8) has two linearly independent solutions satisfyingU±(x) = e±iΘx 1 +O(1/x)
as x→+∞. If we now return to the original variables, we conclude that Eq. (3.2) has two linearly independent solutions φ± such that
φ±(r) = 1
√re±iΘ/r
1 +O(r)
, asr →0. (3.9)
As is easily verified, no nontrivial linear combination ofφ+ and φ−can belong to H1(R+, rdr), which means that Eq. (3.2) has no nontrivial solution if a= 0 and b= 1.
The situation is completely different whenb≤0.
Lemma 3.3. For any m6= 0and k6= 0, the set of all b≤0 such that Eq. (3.2) witha= 0 has a nontrivial solution in H1(R+, rdr) is finite. Moreover :
1) This set is nonempty for a finite number of values of m only;
2) For both the Kaufmann-Scully vortex (1.8) and the Lamb-Oseen vortex (1.9), Eq.(3.2) has no nontrivial solution when a= 0 andb≤0 if |m| ≥2.
Proof. If a= 0 andb≤0, then γ⋆(r) = Ω(r)−b= Ω(r) +|b|>0. In this region, it is easy to verify that the coefficient B(r)<1 defined in (3.3) is an increasing function of both parameters
|m| and |b|. Moreover, using the bounds on Ω, W, and Φ which follow from assumptions H1, H2, see Remark 1.2, we obtain the following estimate :
sup
r>0
1− B(r)
≤ C m2 sup
r>0
Φ(r)
(Ω(r) +|b|)2 +W(r) +r|W′(r)| Ω(r) +|b|
≤ C m2
1 1 +|b|,
where the constant C depends only on the vorticity profile. As a consequence, we see that B(r) ≥ 0 when |m| or |b| is large enough, and this implies that Eq. (3.2) has no nontrivial solution, see (3.10) below. It follows that the linearized operator Lm,k can have eigenvalues s = m(a−ib) with a = 0 and b ≤ 0 only for a finite number of values of m ∈ Z, and using Sturm-Liouville theory we also conclude that, for any m ∈ Z, there exist only finitely many eigenvalues with a= 0 and b ≤0. Interestingly enough, for both the Kaufmann-Scully vortex (1.8) and the Lamb-Oseen vortex (1.9), an explicit calculation which is reproduced in Section 6.7 shows thatB(r)≥1−4/m2, so that there are no eigenvalues in this region when|m| ≥2.
As a final comment, we mention that, whenm=±1, there are always eigenvalues witha= 0 and b≤0. Indeed, due to translation invariance, the operator Lm,0 has a zero eigenvalue with eigenfunction
u = −imΩer+ (W−Ω)eθ, ω = W′ez.
That eigenvalue bifurcates out of the essential spectrum as the parameterkvaries, so thatLm,k has at least one eigenvalue s=−imbwith b <0 if |m|= 1 and |k|is small enough.
3.3 Eigenvalues outside the imaginary axis: Howard identities
For our next step in the study of the eigenvalue equation (3.2), we use a classical method originally due to Rayleigh [20] to show that the linearized operator Lm,k has no spectrum in large regions of the complex plane, which are depicted in Fig. 1. The idea is to derive integral identities satisfied by the hypothetical eigenfunctions, which eventually lead to a contradiction.
Assume thus that the eigenvalue equation (3.2) has a nontrivial solution ur∈H1(R+, rdr), for somes=m(a−ib)∈C, wherea6= 0. Multiplying both sides of (3.2) byr¯urand integrating over R+, we easily obtain, using the results of Section 3.1 :
Z ∞
0
A(r)|∂r∗ur|2+B(r)|ur|2
rdr = 0. (3.10)
Note that the functionBis complex-valued ifa6= 0, so that (3.10) gives two integral relations for the radial velocityur. For instance, taking the imaginary part of (3.10) and using the expression (3.3) of B, we obtain the identity
a Z ∞
0
2(b−Ω(r)) (a2+ (Ω−b)2)2
k2
m2A(r)Φ(r) + r
a2+ (Ω−b)2∂r W(r) m2+k2r2
|ur|2rdr = 0. (3.11) This relation is identically satisfied if a= 0, but gives useful information ifa6= 0. For instance, if b≤0, thenb−Ω(r)<0 for allr >0, and assumption H1 implies that
Φ(r) > 0 and ∂r W(r) m2+k2r2
< 0, for all r >0.
Thus the integrand in (3.11) is nonpositive and not identically zero, hence equality (3.11) cannot hold. We conclude that the operatorLm,k has no eigenvalues=m(a−ib) witha6= 0 andb≤0, see Fig. 1. Unfortunately, we do not know how to use the relation (3.10) to preclude the existence of eigenvalues of Lm,k in other regions of the complex plane.
The following approach, due to Howard [13, 14], provides other identities similar to (3.10), which give further information on the possible eigenvalues. Define ur = q(r)vr, where q is a (real or complex valued) weight function satisfyingq(r)6= 0 for all r >0. Thenvr is a solution to
−∂r
q(r)2A(r)∂r∗vr
+E(r)vr = 0, (3.12)
where
E(r) = q(r)2B(r)−q(r)q′(r)
A′(r)−A(r) r
−q(r)q′′(r)A(r). Multiplying both sides of (3.12) by r¯vr and integrating over R+, we deduce
Z ∞
0
q(r)2A(r)|∂r∗vr|2+E(r)|vr|2
rdr = 0. (3.13)
Ifq is real-valued, thenq2|vr|2=|ur|2 and taking the imaginary part of (3.13) we recover (3.11), but the real part gives new information. Ifq is complex, both the real and the imaginary parts of (3.13) provide new information.
Following [14], we now consider in more detail some interesting particular cases of (3.13).
Choice 1 : q(r) =γ⋆(r). We then have E(r) = γ⋆(r)2− k2
m2 A(r)Φ(r) +rγ⋆(r)∂r
W(r) m2+k2r2
−γ⋆(r)γ⋆′(r)
A′(r)−A(r) r
−γ⋆(r)γ⋆′′(r)A(r). Since rγ⋆′(r) =rΩ′(r) =W(r)−2Ω(r), we observe that
γ⋆′(r)
A′(r)−A(r) r
+γ⋆′′(r)A(r) = r∂rγ⋆′(r)A(r) r
= r∂rW(r)−2Ω(r) m2+k2r2
, (3.14)
